Basic bluff-body aerodynamics II Wind loading and structural response Lecture 9 Dr. J.D. Holmes.

Basic bluff-body aerodynamics II

Wind loading and structural response

Lecture 9 Dr. J.D. Holmes

Basic bluff-body aerodynamics• Pressures on prisms in turbulent boundary layer :

• drag coefficient (based on Uh ) 0.8

-0.20 -0.10 -0.20

-0.23 -0.18 -0.23xx x

-0.20 -0.20x x

xx x

Sym.aboutCL

-0.2

-0.5

-0.8

-0.8

-0.5

-0.8

-0.6

-0.7

0.7

0.50.0

Wind

windward wall

side wall

roofleeward wall

Basic bluff-body aerodynamics

• Pressures on prisms in turbulent boundary layer :

-0.5

-0.4 to –0.49

Leeward wall

-0.5

-0.5

x -0.6

x -0.6

x -0.6

-0.5

-0.6

-0.6-0.7

Wind

Side wall

x 0.4 0.3 x

0.9 x

0.5 x

Windward wall

-0.6

-0.56 to –0.59

-0.6x x

Wind

Roof

shows effect of velocity profile

nearly uniform


• Circular cylinders :

Complexity due to interacting effects of surface roughness, Reynolds Number and turbulence in the approach flow

Flow regimes in smooth flow :

Re < 2 105

Cd = 1.2

Sub-critical

Laminar boundary layer Separation

Subcritical regime : most wind-tunnel tests - separation at about 90o from the windward generator





Supercritical : flow in boundary layer becomes turbulent - separation at 140o - minimum drag coefficient

Re 5 105

Cd 0.4

Super-critical

Laminar Turbulent Separation





Post-critical : flow in boundary layer is turbulent - separation at about 120o

Re 107

Cd 0.7

Post-critical

Turbulent Separation



Pressure distributions at sub-critical and super-critical Reynolds Numbers

20 60 100 140

1.0

0.5

0

-0.5

-1.0

-1.5

-2.0

-2.5

U

degrees

Cp

Drag coefficient mainly determined by pressure on leeward side (wake)



Effect of surface roughness :

Increasing surface roughness : decreases critical Re - increases minimum Cd

1.2

0.8

0.4

U b

104 2 4 8 105 2 4 8 106 2 4 8 107

k/b = 0.02

k/b = 0.007

k/b = 0.002

Sanded surfaceSmooth surface

Cd

Re

increasing surface roughness



Effect of aspect ratio on mean pressure distribution :

Silos, tanks in atmospheric boundary layer

-2

-1.5

-1

-0.5

0

0.5

1

0 90 180

Angle (degrees)

h/b = 0.5

h/b = 1.0

h/b = 2.0

Cp

b

h

Decreasing h/b : increases minimum Cp (less negative)


• Fluctuating forces and pressures on bluff bodies :

Sources of fluctuating pressures and forces :

• Freestream turbulence (buffeting)

- associated with flow fluctuations in the approach flow

• Vortex-shedding (wake-induced)

- unsteady flow generated by the bluff body itself

• Aeroelastic forces

- forces due to the movement of the body (e.g. aerodynamic damping)


• Buffeting - the Quasi-steady assumption :

Fluctuating pressure on the body is assumed to follow the variations in wind velocity in the approach flow :

p(t) = Cpo (1/2) a [U(t)]2

Cpo is a quasi-steady pressure coefficient

Expanding :p(t) = Cpo (1/2) a [U + u(t) ]2 = Cpo (1/2) a [U2 + 2U u(t) + u(t)2 ]

Taking mean values :

p = Cpo (1/2) a [U2 + u2]


• Buffeting - the Quasi-steady assumption :

Small turbulence intensities :

p Cpo (1/2) aU2 =Cp (1/2) aU2

i.e. Cpo is approximately equal to Cp

Fluctuating component :

p' (t) = Cpo (1/2) a [2U u'(t) + u'(t)2 ]

(e.g. for Iu = 0.15, u2 = 0.0225U2 )

Squaring and taking mean values :

Cp2 (1/4) a

2 [4U2 ]= Cp2 a

2 U2 u2 2p 2u


• Peak pressures by the Quasi-steady assumption :

Quasi-steady assumption gives predictions of either maximum

or minimum pressure, depending on sign of Cp

Time

p(t)

p

p

]U[(1/2)ρC]U[(1/2)ρCpor p 2ap

2apo


• Vortex shedding :

On a long (two-dimensional) bluff body, the rolling up of separating shear layers generates vortices on each side alternately

• Occurs in smooth or turbulent approach flow

• may be enhanced by vibration of the body (‘lock-in’)

• cross-wind force produced as each vortex is shed


• Vortex shedding :

Strouhal Number - non dimensional vortex shedding frequency, ns :

• b = cross-wind dimension of body

• St varies with shape of cross section

U

bnSt s

• circular cylinder : varies with Reynolds Number


• Vortex shedding - circular cylinder :

• vortex shedding not regular in the super-critical Reynolds Number range


• Vortex shedding - other cross-sections :

0.08

2b

2.5b

~10b

0.12

0.06

0.14


• fluctuating pressure coefficient :

• fluctuating sectional force coefficient :

2a

2

p

Uρ21

pC

bUρ21

fC

2a

2

f

• fluctuating (total) force coefficient :

AUρ21

FC

2a

2

F


• fluctuating cross-wind sectional force coefficient for circular cylinder :

dependecy on Reynolds Number

105 106 107

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Flu

ctua

ting

side

forc

e co

effic

ient

C

Reynolds number, Re


• Quasi-steady fluctuating pressure coefficient :

• Quasi-steady drag coefficient :

up2

a

2ap

2a

2

p IC2Uρ

21

uUρC

Uρ21

pC

uDD ICC 2


• Correlation coefficient for fluctuating forces on a two-dimensional body :

• Correlation length :

2f

21

2

21

σ

(t)f(t)f

f

(t)f(t)fρ

dyy

0

)(

y is separation distance between sections


• Correlation length for a stationary circular cylinder (smooth flow) :

cross-wind vibration at same frequency as vortex shedding increases correlation length

6

4

2

0

104 105 106

Reynolds number, Re

Correlation length / diameter


• Total fluctuating force on a slender body :

We require the total mean and fluctuating forces on the whole body

L

iii fff

Nf

1f

jf

δy1 δyi δy jδyN



mean total force : F = fi yi i

L

0

i dyf

instantaneous total fluctuating force : F(t) = fi (t) yi

= f1 (t) y1 + f2 (t) y2 + ……………….fN (t) yN

Squaring both sides : [F(t)]2 = [ f1 (t) y1 + f2 (t) y2 + ……………….fN (t) yN]2

= [f1 (t) y1]2 + [f2 (t) y2]2 ..+ [fN (t) yN]2 + f1 (t) f2(t) y1y2 + f1 (t) f3(t) y1y3 +...

jij

N

ji

N

i

δyδy(t)f(t)f



Taking mean values :

As yi, yj tend to zero :

writing the integrand (covariance) as :

ji

N

jji

N

i

yytftfF )()(2

jiji

LL

dydytftfF )()(00

2

)()()( 2jiji yyftftf

ji

L

ji

L

dydyyytfF 00

22 )()(

This relates the total mean square fluctuating force to the sectional force



Introduce a new variable (yi - yj) :

Special case (1) - full correlation, (yi-yj) = 1 :

fluctuating forces treated like static forces

mean square fluctuating force is proportional to the correlation length - applicable to slender towers

)yd(y)yρ(ydyfF ji

yL

y-

ji

L

0

j22

j

j

222 L(t)fF

Special case (2) - low correlation, correlation length is much less than L :

2)yd(y)yρ(y)yd(y)yρ(y ji

-

jiji

yL

y-

ji

j

j

L.2(t)fF 22



Symmetric about diagonal since (yj-yi) = (yi-yj ). Along the diagonal, the height is 1.0

The double integral : is represented by the volume under the graph :

ji

L

0

ji

L

0

dydy)yρ(y

On lines parallel to the diagonal, height is constant

0

0.5

1

1

S1

yi

yj



Consider the contribution from the slice as shown :

Length of slice = (L-z)2

z/2

z /2

L

yi-yj=0

yi-yj= z

yj

yi

Volume under slice = (z)(L-z)22

δz

Total volume =

L

0dzz)ρ(z)(L2

L

0

22 dz z)ρ(z)(L2.fF

(reduced to single integral)

End of Lecture 9

John Holmes225-405-3789 [email protected]

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Basic bluff-body aerodynamics II Wind loading and structural response Lecture 9 Dr. J.D. Holmes.

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